Salvato in:
| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2512.17487 |
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Sommario:
- In this article, we study the algebraic and dynamical structure of certain normal subgroups of the quasi-isometry group of Euclidean space $QI(\mathbb{R}^n)$. For \[ H = \Big\{ [f] \in QI(\mathbb{R}^n) : \lim_{\|x\|\to\infty} \frac{\|f(x)-x\|}{\|x\|} = 0 \Big\}, \] and for $0<α<1$, \[ H_α= \Big\{ [f] \in QI(\mathbb{R}^n) : \|f(x)-x\| \le K\|x\|^α\text{ for large } \|x\| \Big\}, \] we show that each $H_α$ is a nontrivial normal subgroup of $QI(\mathbb{R}^n)$, satisfying \[ H_α\subset H_β\subset H \qquad \text{for } 0<α<β<1. \] We prove that the centers of $QI(\mathbb{R}^n)/H$ and $QI(\mathbb{R}^n)/H_α$ are trivial, while these quotients admit nontrivial torsion elements. Consequently, they are neither left-orderable nor locally indicable. Finally, we introduce an asymptotic topology on $QI(\mathbb{R}^n)$ and show that the family $\{H_α\}_{0<α<1}$ is dense in $H$.